Multivariate Distributions - CMU Statistics

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chapter 14Multivariate Review of DefinitionsLet s review some definitions from basic probability . When we have a random vector Xwithpdifferent components,X1,X2,...Xp, thejoint cumulative distributionfunctionisF( a)=F(a1,a2,...ap)=Pr X1 a1,X2 a2,...Xp ap ( )ThusF( b) F( a)=Pr a1<X1 b1,a2<X2 b2,...ap<Xp bp ( )This is the probability thatXis in a (hyper-)rectangle, rather than just in an probability density functionisp( x)=p(x1,x2,...xp)= pF(a1,...ap) ap a= x( )Of course,F( a)= a1 a2 ... ap p(x1,x2,...xp) ( )(In this case, the order of integration doesn t matter. Why?)From these, and especially from the joint PDF, we can recover the marginal PDFof any group of variables, say those numbered 1 throughq,p(x1,x2.)

Chapter 14 Multivariate Distributions 14.1 Review of Definitions ... the probability density of the multivariate Gaussian is p ... 14.2.3 Projections of Multivariate Gaussians A useful fact about multivariate Gaussians is that all their univariate projections are alsoGaussian.

  Chapter, Distribution, Probability, Multivariate, Multivariate distributions